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行列式:
) R& `. W+ m, {# d; j %Aij = (-1)^(i+j)*Mij6 O' B* w9 o$ \6 o
%D = a11*A11 + a12*A12 + a13*A13 + ... + a1n*A1n
+ G4 M& m9 z) d2 `# z%H = [1 2 3;7 9 7;5 6 1;];det(H) = 14) C' N, q7 C! }7 [! v2 @
function H_Det = Matrix_Det(H,N) %输入矩阵 阶数
2 O1 V1 Z& g1 G+ c2 U0 [& F) PH_Det = 0;0 J& Z# ^% E8 d1 t; t& [4 i
if N==1
U' w. x; ^$ ]0 e0 A, `1 e! E H_Det = H(1,1); 4 ?8 z% p1 Z* ^
return;
( I2 ~/ H3 I: d) Gend2 Z4 _% o8 D' z. ~2 r' @+ N' E
temp = zeros(N-1,N-1);5 E3 x. }$ T% D; N8 I$ O5 f
for i=1:N
5 W, n7 M9 ?7 l5 U% M( J for j=2:N %第二行开始# @/ F* _6 x* x( y# M+ O
for k=1:N-1* d0 y# |* C. q& n' u9 s
if k>=i
7 Z& F4 r4 e; Z2 \7 C cln = k+1;* Y: u, `; B, r( d% C/ [" t# d
else
; I" g7 q: r9 F) j$ X2 q% \0 ^! L cln = k;
! @0 B& `% i9 I8 [$ F end
' g1 r0 a6 k+ g7 a7 i% ?7 C temp(j-1,k) = H(j,cln);9 J0 q7 `$ m9 J7 ]! b
end
4 r3 u( s' k# A& w( O end
: O8 T$ Q4 i0 {! I t = Matrix_Det(temp,N-1); %递归
9 q) g% u2 }4 @/ n# C( x* \4 ]; q if rem(1+i,2)==0 %(-1)^(1+i)1 ?& p8 g, N) G
H_Det = H_Det+H(1,i)*t;
+ e+ `0 F5 [. q. }* m w else
4 V2 v+ t6 y. |/ k" ~* v H_Det = H_Det-H(1,i)*t;
) A# N& N w3 F3 w+ V end' I V* e" n4 `0 s J; B" c0 x
end. P) b! x# J: T& g7 O u2 Q
+ B1 U- V4 y( \) r) G% n$ uend8 ?! ^5 h$ S! n. M
1 S! R* ]5 r! R( z6 `$ L伴随阵:
6 L' P# n. g, I6 {: `. C- o%伴随阵A* = |Aij|(nxn)* d! o9 L3 P+ c) i( y- s( J' g
%Aij = (-1)^(i+j)*Mij
$ M7 v6 V2 j5 I# [; c$ g%H = [1 2 3;7 9 7;5 6 1;];det(H)*inv(H) = [-33 16 -13;28 -14 14;-3 4 -5];' G' S/ Z( Q9 P2 D
function H_Adjoint = Matrix_Adjoint(H,N)% f4 G1 h% B# t1 a8 W
H_Adjoint = zeros(N,N);& ^/ I$ B' D# M% K1 e% q
if N==1
+ E* |' S5 v0 g9 x! \% H_Adjoint = [H(2,2) H(2,1);H(1,2) H(1,1)];
]/ m. Z, |+ y6 E* ?/ V. v H_Adjoint = 1;
2 t* V7 C9 v7 {8 ?: X return;
" @7 ^* B0 w0 F1 ^4 }+ Kend9 l6 N' L( w. u$ o o
temp = zeros(N-1,N-1);5 L. k+ J6 A5 M9 R" i% G
for i=1:N %逐行扫描
0 O9 @0 B$ o6 b7 f8 w for j=1:N %逐列扫描
7 W0 k( }, V J& Z+ q2 |: V for k=1:N-1 %行组合" l1 _' p( T0 s0 a+ G! x2 ^0 m. M
if k>=j
$ j/ {) z8 j7 F2 Z* | row = k+1;& @3 g; [. J7 j+ }
else; D" P! r3 D: u4 @- Z
row = k;1 c; J, v, H" N$ H/ H5 A! ^
end/ H, |1 l- Q j2 Q' t5 J1 U
for t=1:N-1 %列组合 * O' @ f* {) R" q6 H. k+ B
if t>=i5 v3 e" t# m5 p
cln = t+1;
1 h4 N" A/ c! r/ Y7 R0 C2 R- L% a else
( M2 G4 I: [: N' p d cln = t;: M( y" V' b- R7 ~& s7 h6 X
end
' Y: E D. |- C$ }2 I0 I temp(k,t) = H(row,cln);
8 y7 s! r% V* U end
{9 E k% w+ i2 Z end
+ S. R# U; q* l8 A/ r6 ?' Q5 D if rem(i+j,2)==0
& R( ~. }$ j" r) s H_Adjoint(i,j) = Matrix_Det(temp,N-1); %求代数余子式! }: F. a4 h; a, K' C
else
* ~; j* ]5 ^3 A# q H_Adjoint(i,j) = -Matrix_Det(temp,N-1);" E3 z0 }! g7 `% F) g
end
- ?3 z7 Q( u$ v/ h9 r2 G end4 |1 i; `- ^( ` h$ p
end
6 f: X' P- E9 P/ zend# q/ W! g( M" F3 w; y" t
# S5 y$ B0 X/ x/ d* v
逆:# ]/ |9 K+ m# O: m! b* J8 H
%A^(-1) = 1/|A|*(A*)( W$ _( [. ?7 X
%H = [1 2 3;7 9 7;5 6 1;];inv(H) = [-2.3571 1.1429 -0.9286;2.0000 -1.0000 1.0000;-0.2143 0.2857 -0.3571];
- T' T/ _$ }, G% p6 O" tfunction H_Inv = Matrix_Inverse(H,N)
& t, Z1 o5 k6 r6 {/ xH_Det = Matrix_Det(H,N);
& f* D9 g" v) K2 @) bH_Adjoint = Matrix_Adjoint(H,N);5 O' L m$ R z1 u
H_Inv = H_Adjoint/H_Det;3 \' _8 L2 O% y2 j3 L
end1 N3 m8 R4 H- o. L
0 S1 }0 Z3 V D' P5 Z
4 @* R q' z, M5 P9 i7 w! k: G8 O1 Q. P" O- |1 L9 a
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